# Problem with equicontinuous and uniformly bounded.

Here is the problem I'm trying to prove :

Let $F$ be a subset of $C([a,b])$. If each sequence in $F$ contains a uniformly convergent sequence, then $F$ is both equicontinuous and uniformly bounded.

As the assumption said, I let {$f_n$} be a sequence in $F$, so that there exists a subsequence {$f_{1,_k}$} which converges uniformly.

Since {${f_n}$} $\setminus$ {$f_{n_k}$} is still a sequence in $F$, so there also exists a uniformly convergent subsequence
{$f_{2,_k}$} in it.

Continue this process, I get a family of sequence {$f_{t,_k}$} with $\bigcup _t$ {$f_{t,_k}$} = {$f_n$}, and each {$f_{t,_k}$} is uniformly convergent.

Now I want to prove the result by showing that any {$f_n$} is equicontinuous and uniformly bounded, then $F$ will be. So I foucus on analyzing each {$f_{t,_k}$}. But evey method I've tried was stopped because of lacking of condition.

I don't know what I should do to keep going.

• Have you already seen the proof of Arzela-Ascoli (which is essentially the converse of this statement)? – Ian Oct 5 '17 at 2:52
• @Ian Yes, I have. I followed how it was proved by writhing each subsequence down, and tried to use diagonal process. But I can't find out any relation between two different {$f_{t,k}$}. – Hung Oct 5 '17 at 3:01

Hint: In fact $F \subseteq C[a,b]$ is a compact set under sup-norm so it is bounded which in language of functions means uniformly bounded.